A vector is a quantity in mathematics that has a magnitude (distance, velocity, or size) and a direction (as on a compass needle, like west, up, southeast, down, or north by northwest). Unlike scalars, which are quantities that have only magnitude (such as temperature, mass, or speed), vectors represent quantities like force, velocity, or displacement that require both a numerical value and a direction to be fully described.

Vectors are commonly represented in two ways:

Geometrically: as arrows, where the length represents the magnitude and the arrow points in the direction
Algebraically: using components, such as $\text{[math]}$ in two dimensions, where $\text{[math]}$ and $\text{[math]}$ are the horizontal and vertical components respectively

Now that we understand the definition of a vector, we will proceed to study some of the basic operations that can be performed between vectors and how to label them.

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Vector Symbol

To represent vectors, mathematicians, physicists, and engineers use rays, labeling them with lowercase or uppercase letters, like this:

Tips for Labeling Vectors

All vectors are named tail (starting point) to arrowhead, which is why we have vector AB and not vector BA.
If you are labeling your vectors, the decision to use uppercase or lowercase lettering rests with you; if you are given vectors, pay attention to the direction of the vector (look at the arrowheads).
Vectors can be parallel and pointing in the same or opposite directions (look at the arrowheads).
Vectors of equal magnitude but pointing in opposite directions are opposites, so vector b can also be written as -a, which negates the magnitude of a.

A vector with unit length is called a unit vector and is symbolized by what is known as a hat: ˆ

Vector Addition

If we have two vectors $\text{[math]}$ and $\text{[math]}$ , then the sum of $\text{[math]}$ and $\text{[math]}$ is

$\text{[math]}$

In other words, the sum vector of $\text{[math]}$ and $\text{[math]}$ is the vector that results from adding the respective components of these vectors: the first component of $\text{[math]}$ is added to the first component of $\text{[math]}$ , and the second component of $\text{[math]}$ is added to the second component of $\text{[math]}$ .

Graphic Interpretation of Addition

Let's observe the following graph that shows the addition of vectors $\text{[math]}$ and $\text{[math]}$ :

representación gráfica de la suma de dos vectores u y v

If $\text{[math]}$ and $\text{[math]}$ are two free vectors, then to add them graphically we first choose the representative of $\text{[math]}$ whose origin is the endpoint of $\text{[math]}$ . Then, $\text{[math]}$ is the vector whose origin is the origin of $\text{[math]}$ and whose endpoint is the endpoint of $\text{[math]}$ .

Note that we can also choose a representative of $\text{[math]}$ such that its origin is the endpoint of $\text{[math]}$ . The sum $\text{[math]}$ will have the same value, but now we will obtain it by connecting the origin of $\text{[math]}$ with the endpoint of $\text{[math]}$ .

Parallelogram Rule

What we discussed earlier as the graphic addition of vectors is known as the parallelogram rule. In particular, if we want to add two free vectors with a common origin, then we must draw lines parallel to the vectors. In this way, a parallelogram is obtained whose diagonal (which starts at the origin of the vectors) is the sum of the vectors itself.

Observe the following figure that shows the parallelogram rule.

regla del paralelogramo representacion grafica con los vectores u y v

Vector Subtraction

The subtraction of two vectors $\text{[math]}$ and $\text{[math]}$ is simply the addition of $\text{[math]}$ with $\text{[math]}$ (that is, the opposite of $\text{[math]}$ ).

Thus, if we consider the components of $\text{[math]}$ and $\text{[math]}$ , then the subtraction is given by:

$\text{[math]}$

Graphically, the subtraction of $\text{[math]}$ and $\text{[math]}$ is obtained the same way as addition. The only difference is that we add the opposite of $\text{[math]}$ . Observe the following figure that shows $\text{[math]}$ and note that at the endpoint of $\text{[math]}$ we place the origin of $\text{[math]}$ .

resta de u y v representacion grafica vectores

Note that graphically, the subtraction vector $\text{[math]}$ connects the endpoint of $\text{[math]}$ with the endpoint of $\text{[math]}$ .

Scalar Multiplication

The multiplication of a vector $\text{[math]}$ by a number $\text{[math]}$ is written $\text{[math]}$ or $\text{[math]}$ . The number $\text{[math]}$ is also known as a scalar. Furthermore, scalar multiplication is another vector that satisfies the following properties:

$\text{[math]}$ has the same direction as $\text{[math]}$ .
If $\text{[math]}$ is positive, then $\text{[math]}$ has the same sense as $\text{[math]}$ .
If $\text{[math]}$ is negative, then $\text{[math]}$ has the opposite sense to $\text{[math]}$ .
The magnitude of $\text{[math]}$ is $\text{[math]}$

Observe the following figure that represents the multiplication of $\text{[math]}$ by 3.

multiplicacion de un vector u por 3 representacion grafica

In terms of components, if $\text{[math]}$ , then scalar multiplication is given by:

$\text{[math]}$

Magnitude of a Vector

The magnitude of a vector is shown as an absolute value, |a|, or with two lines to avoid confusing it with absolute value, ||a||.

If you know the x-axis and y-axis values of a vector (as if it were on a map or cartesian coordinate system) you can easily calculate its magnitude by applying the Pythagorean Theorem to the change in position from tail to arrowhead:

So here, with the tail at (1, 4) and the arrowhead at (7, 8), we have a change in x-value of 6 and a change in y-value of 4, so:

$∥ u ∥ = {\sqrt{6}}^{2} + 4^{2}$ ∥u∥=62+42

$∥ u ∥ = \sqrt{36} + 16$ ∥u∥=36+16

$∥ u ∥ = \sqrt{52}$ ∥u∥=52

$∥ u ∥ = 7.2111 u n i t s$ ∥u∥=7.2111units

The magnitude of the vector is 7.2111 units.

The unit is determined by what you are measuring; inches, kilometers, miles per hour (mph), etc. So, if we just measured distance in miles, then 7.2111 miles would be the length of the vector.

Scalar vs. Vectors

To be clear, scalar quantities are magnitude only: mass, temperature, speed, volume, distance, energy, work, and so forth. Think of them as pure numbers.

Displacement Vectors

Flying superheroes seldom take the shortest route from Daily Bugle or Daily Planet to that day's disaster. They swoop, loop, leap and barrel roll before finally arriving in the nick of time.

If we used vectors to chart a flying superhero's course, we might need five or six vectors to take into account all those detours. A displacement vector cuts from start to finish in a straight line:

Displacement in this meaning comes from physics, meaning the change in position from the initial position.

You put your right hand in; you take your right hand out: Zero displacement. You cha-cha three steps to the left and two steps to the right: A displacement vector of one step left.

In this figure, we can see the displacement vector is also the resultant.

The calculation of displacement is still vector n + vector v = r, because vectors a and y negate each other!

You may think we wasted a lot of effort to move so small a distance, but what if we were in a Navy vessel and had to navigate around a jetty or protected wildlife preserve? Then we can see this really was the shortest route.

Vector Examples

All of these measurements are examples of vectors because they all involve distance or size of a force and a direction:

Velocity
Force
Acceleration
Momentum
Displacement

Commercial airliners, fighter jets, boats, cars, bicyclists, runners, falling objects, rockets, hot air balloons, paper airplanes, and submarines are all examples of moving objects that use vectors in everyday life.

Pilots and navigators must use vectors to get to their destinations. Rocket scientists and aerospace engineers use vectors to control rockets.

There is one exception to vectors having length and direction, and that is the zero vector. The zero vector has no length, so it is not pointing in any particular direction. This means the zero vector has an undefined direction.

Basic Vector Problems

What will happen if we multiply a vector by 4? We hope you said it will point in the same direction but will be four times longer!
Is "25 knots south by southwest" a scalar or a vector? That is a vector, since it gives a magnitude and direction.
What will happen to a vector if we multiply it by $- \frac{1}{2}$ −21? We hope you said it would be half as long and going in the opposite direction!
Two vectors are parallel but pointing in opposite directions. One is vector z. What is the other vector? We hope you remembered about negating vectors, by labeling it -z

Example Exercises with Vectors

Consider the vectors $\text{[math]}$ and $\text{[math]}$ . Thus:

1 The sum is given by:

$\text{[math]}$

2 The subtraction is:

$\text{[math]}$

3 The opposite of $\text{[math]}$ is:

$\text{[math]}$

4 The scalar product of $\text{[math]}$ by 3 is given by:

$\text{[math]}$

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.

Vector Operations